Why full spin polarization suppresses the resistivity-peak temperature $T_{\max}$ in the clean two-valley two-dimensional metal-insulator transition

In ultra-high-mobility, two-valley SiGe/Si/SiGe quantum wells, the resistivity on the metallic side of the zero-field two-dimensional metal-insulator transition (2D MIT) peaks at a temperature $T_{\max}$ that closely tracks the renormalized Fermi temperature $T_{\mathrm{F}}$. When the electron spins are fully polarized by a parallel magnetic field, $T_{\mathrm{F}}$ approximately doubles, yet $T_{\max}$ decreases appreciably. Dynamical mean-field theory (DMFT), which predicts $T_{\max}\sim T_{\mathrm{F}}$, then gets the wrong sign of the shift, while renormalization-group (RG) scaling theory predicts a decrease but fails at zero field and is restricted to the diffusive, low-resistivity regime; meanwhile both spin states show the same universal DMFT scaling of the low-temperature resistivity drop. The problem is to identify the microscopic mechanism and construct a quantitative theory that predicts the observed spin dependence of $T_{\max}$ in a clean, strongly interacting two-valley 2D electron system — treating spin polarization, intervalley scattering, and residual disorder across the diffusive-to-ballistic crossover — or to demonstrate experimentally which candidate mechanism is at work.

Background

Two-dimensional (2D) electron systems in semiconductors host a long-standing puzzle of condensed-matter physics: the apparent metal-insulator transition (MIT) at zero magnetic field. The scaling theory of localization predicts that non-interacting 2D electrons are always localized, yet in 1994 a density-driven transition between metallic and insulating behavior was observed in a 2D electron gas in silicon metal-oxide-semiconductor field-effect transistors (Si MOSFETs), the devices in which a gate controls the carrier density (Kravchenko et al., Physical Review B 50, 8039 (1994)). The transition is now widely attributed to strong Coulomb interactions, quantified by the interaction parameter $r_s$, the ratio of the typical Coulomb energy to the Fermi energy, which grows as the carrier density is lowered.

In silicon-based 2D systems the conduction band has two nearly degenerate energy minima, called valleys, which act as an additional twofold flavor degeneracy ($g_v=2$); valley degeneracy enhances correlation effects and is expected to promote metallicity. On the metallic side near the transition, the resistivity $\rho(T)$ is non-monotonic: as the temperature $T$ is lowered, $\rho$ first rises, reaches a maximum at a temperature $T_{\max}$, and then falls — in the cleanest samples by more than an order of magnitude. Two frameworks describe this maximum from opposite limits. Renormalization-group (RG) scaling theory of disordered, interacting electrons (Finkel'stein's approach, extended to multivalley systems by Punnoose and Finkel'stein, Physical Review Letters 88, 016802 (2001)) treats the interplay of disorder and interactions and is controlled only in the diffusive regime, roughly $k_{\mathrm{B}}T\tau/\hbar<1$ (where $\tau$ is the elastic mean-free time between collisions), at resistivities well below $\pi h/e^2$. Dynamical mean-field theory (DMFT), a non-perturbative method that solves the Hubbard model of interacting electrons on a lattice with a local, self-consistently determined self-energy, treats interactions as the driving force of a disorder-assisted Wigner-Mott transition — an insulating state interpolating between a Wigner crystal, the electron solid stabilized by pure Coulomb repulsion, and a Mott insulator, where localization is driven by the filling of the band (Camjayi et al., Nature Physics 4, 932 (2008); Radonjić et al., Physical Review B 85, 085133 (2012)); it predicts that $T_{\max}$ is the quasiparticle coherence temperature, of order the Fermi temperature $T_{\mathrm{F}}$ renormalized by interactions, and that the normalized resistivity $[\rho(T)-\rho(0)]/[\rho_{\max}-\rho(0)]$ is a universal function of $T/T_{\max}$.

$T_{\mathrm{F}}$ is renormalized through the strongly enhanced effective mass $m$ measured from the density of states. A parallel magnetic field $B^{*}$ large enough to saturate the magnetoresistance fully polarizes the spins, lifting the spin degeneracy from $g_s=2$ to $g_s=1$ and, at fixed density and unchanged mass, approximately doubling $T_{\mathrm{F}}$; equivalently, for fully polarized electrons $k_{\mathrm{B}}T_{\mathrm{F}}(B^{*})=g_{\mathrm{F}}\mu_{\mathrm{B}}B^{*}$ with the $g$-factor $g_{\mathrm{F}}\approx 2$ and $\mu_{\mathrm{B}}$ the Bohr magneton.

The present question arises in ultra-high-mobility SiGe/Si/SiGe quantum wells — a strained silicon layer, which hosts the two valleys, sandwiched between silicon-germanium barriers — whose residual short-range disorder is about two orders of magnitude lower than in Si MOSFETs, so that intervalley scattering (scattering between the two valley minima, which requires short-range potential) is weak and the two valleys remain distinct. There the zero-field MIT occurs at $n_c(0)\approx 0.88\times 10^{10}\ \mathrm{cm}^{-2}$ with a record low-temperature resistivity drop (by a factor of about 12) (Melnikov et al., Physical Review B 99, 081106(R) (2019)), and the metallic state survives full spin polarization with $n_c(B^{*})\approx 1.11\times 10^{10}\ \mathrm{cm}^{-2}$ (Melnikov et al., Physical Review B 101, 045302 (2020)) — in contrast to Si MOSFETs, where full spin polarization completely quenches the metallic temperature dependence of the resistance. The observed ratio $n_c(B^{*})/n_c(0)\approx 1.2$ agrees with calculations of Dolgopolov, Shashkin, and Kravchenko (Physical Review B 96, 075307 (2017)). In these wells, $T_{\max}(0)$ closely tracks $T_{\mathrm{F}}(0)$, and the DMFT universal scaling of the low-temperature resistivity drop holds in both spin-unpolarized and fully spin-polarized states, while the RG scaling collapses only in a narrow density window near the critical density.

The unresolved observation (Shashkin and Kravchenko, Journal of Experimental and Theoretical Physics 135, 432 (2022); first reported in Shashkin et al., Scientific Reports 12, 5080 (2022)) sits between these two successes: under full spin polarization $T_{\max}$ decreases appreciably even though $T_{\mathrm{F}}$ approximately doubles — a conclusion placed on firm ground by the later measurement that the strongly enhanced effective mass is independent of the degree of spin polarization (Melnikov et al., Scientific Reports 13, 17364 (2023)). DMFT therefore predicts the wrong sign of the shift of $T_{\max}$, RG scaling predicts the sign but fails at zero field and lies outside its diffusive-regime validity, and the effect cannot be blamed on disorder-level changes because $T_{\max}\sim T_{\mathrm{F}}$ still holds at zero field in dirtier silicon systems. The authors state that the observed behavior of $T_{\max}$ cannot be described by existing theories, and attribute the effect to the electrons' spins without identifying a mechanism.

Problem Statement

Identify the microscopic mechanism that makes the resistivity-peak temperature $T_{\max}$ of the clean, strongly interacting two-valley 2D electron system spin-dependent — close to the renormalized Fermi temperature $T_{\mathrm{F}}$ in zero magnetic field but appreciably lower under full spin polarization, even though $T_{\mathrm{F}}$ approximately doubles when the spin degeneracy is lifted — and, on that basis, either construct a quantitative theory that predicts $T_{\max}$ as a function of the electron sheet density $n_s$ and the parallel magnetic field $B$ in both spin states, or demonstrate experimentally which mechanism is at work.

The origin of this suppressed coherence/crossover scale under spin polarization is constrained by several linked conditions: (i) a theory must treat the spin-polarized two-valley band structure, weak intervalley scattering, and residual short-range disorder explicitly, and must apply in the ballistic regime (roughly $k_{\mathrm{B}}T\tau/\hbar>1$) where the experiments lie, not only in the diffusive regime where present theories are controlled; (ii) it must remain consistent with the universal DMFT scaling of the low-temperature resistivity drop that holds in both spin states, and say how this coexists with the RG scaling description that is valid only in the diffusive, low-resistivity window; (iii) it must account for the role of intervalley scattering and short-range disorder in whether the metallic state is quenched (Si MOSFETs) or survives (SiGe/Si/SiGe quantum wells) under full spin polarization; (iv) an experimental resolution must discriminate among the candidate mechanisms for the spin dependence of $T_{\max}$ — in particular the Wigner-Mott/DMFT coherence scenario and the other candidate scenarios proposed in the literature, the leading ones of which at present allow a quantitative comparison with the experiment.

The account must be consistent with the observed behavior: $T_{\max}(0)\approx T_{\mathrm{F}}(0)$; a decrease of $T_{\max}$ under full spin polarization while $T_{\mathrm{F}}$ approximately doubles with a spin-independent enhanced effective mass; the universal scaling of $[\rho(T)-\rho(0)]/[\rho_{\max}-\rho(0)]$ versus $T/T_{\max}$ in both spin states; and the quenching or survival of metallicity correlated with intervalley-scattering strength.

References

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